Final answer:
The force required to hold the spring stretched at a distance of 12 cm beyond its natural length is 2/3 N, can be determined using Hooke's Law, which relates the force to the spring's displacement and force constant.
Step-by-step explanation:
The force required to stretch a spring can be calculated using Hooke's Law, which states that the force is proportional to the distance the spring is stretched. The formula for Hooke's Law is F = kx, where F is the force, k is the spring constant, and x is the distance stretched. In this case, the work done to stretch the spring from its natural length to 12 cm beyond its natural length is equal to the change in the potential energy of the spring. The work done is given by the formula W = 1/2kx^2, where W is the work, k is the spring constant, and x is the distance stretched. Since the work done is equal to the change in the potential energy, we can set the work done equal to the change in potential energy and solve for the force:
W = 1/2kx^2
The work done to stretch the spring from its natural length to 12 cm beyond its natural length is four times the work done to stretch it from its natural length to 6 cm beyond its natural length. Therefore, 4W = W + W, or 4W = 2W. So, W = (1/2)k(6^2) = 18k.
The force required to hold the spring stretched at a distance of 12 cm is equal to the spring constant multiplied by the distance stretched. Therefore, the force is F = kx = k(12) = 12k.
Using the given information, we can substitute the value of W into the equation and solve for the force:
12k = 18k
12 = 18
k = 12/18 = rac{2}{3}
So, the force that holds the spring stretched at a distance of 12 cm beyond its natural length is rac{2}{3} N.